# How to memorize the trigonometric identities?

I am stuck trying to memorize the trig identities, and try as I may, I just can't get them to stick (especially the sum-product and product-sum formulas). I am concerned I won't be able to memorize them in time for my test, and I was wondering if there was a better way than rhote memorization.

Any suggestions?

Thanks.

EDIT: Passed the test, thanks to your good suggestions. Thank you all!

• I personally find it easier not to memorize, but to be able to derive. Maybe you don't need to memorize them directly, but some derivation starting point that is easier or more intuitive to remember (as long as you can do the derivation quickly when you need it). Jul 24, 2014 at 17:58
• Under time pressure, this isn't very convenient when proving new identities. A lot of them I can derive; it just takes me a long time (30-40 seconds each time) which is time I just don't have to waste. Jul 24, 2014 at 18:20
• Nov 28, 2018 at 14:10

To be honest, I would always forget all my trig identities until I learned complex numbers. Assuming you don't want to go there, try to be as efficient as possible. I tell my students to remember these at the bare minimum:

$$\cos(A\pm B) = \cos(A)\cos(B)\mp \sin(A)\sin(B)\\ \sin(A\pm B) = \sin(A)\cos(B)\pm \cos(A)\sin(B)$$

Where $\mp$' means to flip the sign i.e. $A+B$ inside becomes $-$ outside. Then, you can get a lot of the other identities by simply adding or subtracting these! For example, the product-to-sum rule for cosine comes from adding the formulae for $\cos(A+B)$ and $\cos(A-B)$, and the product-to-sum rule for sin comes by subtracting them.

In an exam setting, you'll always be more efficient by having things memorized though, so going from scratch' should probably be a last resort.

• My answer here gives a picture of the sum and difference identities ... and a cheer to help keep them in your head. :)
– Blue
Jul 24, 2014 at 18:35

If you're familiar with basic manipulation of complex numbers, one way to remember (or quickly "re-derive") various identities is to use DeMoivre's formula:

$$e^{i\theta} = \cos\theta+i\sin\theta.$$

For instance, you can use this to find a formula for $\sin 2 \theta$ as follows:

$$\sin 2\theta = Im(e^{i(2\theta)}) = Im((e^{i\theta})^2) = Im((\cos \theta+i\sin\theta)^2)=$$

$$Im(cos^2\theta-\sin^2\theta+2i \cos\theta\sin\theta).$$

Taking the imaginary part of this last expression, we see that

$$\sin 2\theta=2\cos\theta\sin\theta.$$

• This looks a lot like DeMoivre's Theorem, which I learned a few weeks ago. What's up with the base, though? And how can you raise a number to an imaginary power? Jul 24, 2014 at 18:21
• You're absolutely correct. This is DeMoivre's, Euler's says $e^{2i\pi}+1=0$. I also messed up the $2\pi$. I'll fix it. Jul 24, 2014 at 18:23

From $$e^{i\theta}=\cos\theta+i\sin\theta$$ and the property of exponential you can find all the trigonometric relation you want.

• although a good way to understand all of their importance, imagine trying to re-derive all of them using this every time. Jul 24, 2014 at 18:01
• Also, most (American) pre-calc level students haven't learned this. Jul 24, 2014 at 18:03
• Sorry, didn't see your comment before making the same suggestion. I often used these formulas to re-derive trig identities during physics tests. Granted, I was well beyond precalc and the tests were hours long. Whether or not they are helpful depends on what sort of test the OP will be taking. Jul 24, 2014 at 18:10

The way that I used to memorize the sum of angle formulas is by memorizing the double angle formulas instead. For example $$\sin 2\theta = 2\sin\theta\cos\theta = \sin\theta\cos\theta+\cos\theta\sin\theta$$ reminds me that $$\sin(A+B) = \sin A\cos B + \cos A\sin B$$ similarly $$\cos2\theta = \cos^2\theta-\sin^2\theta = \cos\theta\cos\theta - \sin\theta\sin\theta$$ reminds me that $$\cos(A+B) = \cos A\cos B - \sin A\sin B$$